\(B=2+2^2+2^3+...+2^{100}\)

\(\Rightarrow2B=2^2+2^3+2^4+...+2^{101}\)

\(\Rightarrow2B-B=2^2+2^3+2^4+...+2^{101}-2-2^2-2^3-...-2^{100}\)

\(\Rightarrow2B-B=2^{101}-2\)

bài 1

2¹⁰¹ - 2

\(A=1+3+3^2+3^3+...+3^{99}\\ \Rightarrow A=\left(1+3+3^2+3^3\right)+\left(3^4+3^5+3^6+3^7\right)+...+\left(3^{96}+3^{97}+3^{98}+3^{99}\right)\)

\(\Rightarrow A=\left(1+3+3^2+3^3\right)+3^4\left(1+3+3^2+3^3\right)+...+3^{96}\left(1+3+3^2+3^3\right)\)

\(\Rightarrow A=\left(1+3+3^2+3^3\right)\left(1+3^4+...+9^{96}\right)\)

\(\Rightarrow A=40\left(1+3^4+...+9^{96}\right)⋮40\)

Ta có : \(1^2+2^2+3^2+.....+10^2=385\)

\(\Leftrightarrow2^2\left(1^2+2^2+3^2+.....+10^2\right)=2^2.385\)

\(\Leftrightarrow2^2+4^2+6^2+.....+20^2=4.385\)

\(\Leftrightarrow2^2+4^2+6^2+.....+20^2=1540\)

Sửa đề: CHo 1²+2²+...+10²=385. Tính S = 2²+4² +...+ 20²

S = 2² + 4² +...+ 20²

= (1.2)² + (2.2)² +...+ (2.10)²

= 1².2² + 2².2² +...+ 2².10²

= 2²(1² + 2² +...+ 10²)

= 4.385

= 1540

S= 1^2  .  2^2 + 2^2 . 2^2  + ... + 10^2 . 2^2
  = 4. ( 1^2 + 2^2 +...+ 10^2)  = 4 . 385 = 1540

chúc bạn học giỏi

\(S=1-2+2^2-2^3+...+2^{2012}-2^{2013}\)

\(\Rightarrow2S=2-2^2+2^3-2^4+...+2^{2013}-2^{2014}\)

\(\Rightarrow2S+S=2-2^2+2^3-...-2^{2014}+1-2^2-2^3+...-2^{2013}\)

\(\Rightarrow3S=1-2^{2014}\)\(\Rightarrow3S-2^{2014}=1-2^{2015}\)

a \(4S=4+4^2+4^3+...+4^{24}\)

\(S=\frac{4S-S}{3}=\frac{4^{24}-1}{3}\)

b/  Xem lại đề bài\(3S=4^{6x}-1=4^{24}-1\Rightarrow6x=24\Rightarrow x=4\)

Ta có : S = 1 - 2 + 2² - 2³ + .... - 2²⁰⁰⁹ + 2²⁰¹⁰

 => 2S = 2 - 2² + 2³ - 2⁴ + .... - 2²⁰¹⁰ + 2²⁰¹¹

Lấy 2S trừ S theo vế ta có : 

2S + S = (2 - 2² + 2³ - 2⁴ + .... - 2²⁰¹⁰ + 2²⁰¹¹) + (1 - 2 + 2² - 2³ + .... - 2²⁰⁰⁹ + 2²⁰¹⁰) 

       3S = 2²⁰¹¹ + 1

Khi đó 3S - 2²⁰¹¹ = 2²⁰¹¹ + 1 - 2²⁰¹¹ = 1  

ccđmvvh

Quá dễ

Đặt A=1 + 2 + 2²+ 2³+ 2⁴ +... + 2⁹⁹ + 2¹⁰⁰

=>2A=2 + 2²+ 2³+ 2⁴ +... + 2⁹⁹ + 2¹⁰⁰+2¹⁰¹

=>2A-A=(2 + 2²+ 2³+ 2⁴ +... + 2⁹⁹ + 2¹⁰⁰+2¹⁰¹)-(1 + 2 + 2²+ 2³+ 2⁴ +... + 2⁹⁹ + 2¹⁰⁰)

=>A=2¹⁰¹-1

\(A=1+2+2^2+2^3+2^4+...+2^{99}+2^{100}\)

\(\Rightarrow2A=2+2^2+2^3+2^4+2^5+...+2^{100}+2^{101}\)

\(\Rightarrow2A-A=2^{101}-1\)

\(\Leftrightarrow A=2^{101}-1\)

Đặt biểu thức là A

ta có 2A-A=2^101-1

Đặt \(A=1+2+2^2+2^3+2^4+...+2^{99}+2^{100}\)

\(\Rightarrow2A=2+2^2+2^3+...+2^{100}+2^{101}\)

\(\Rightarrow A=2A-A=\left(2+2^2+2^3+2^4+...+2^{101}\right)-\left(1+2+2^2+2^3+...+2^{100}\right)=2^{101}-1\)